'When should we book our flight, is it better to book in advance?' The first part of that question is tricky, the second easy. Yes, book in advance. When to book? Difficult to say. Certainly not now, as tickets go on sale from 310 - 330 days before departure. Very early booking leaves one open to schedule changes by airlines. Lowest fares may not be available from earilest booking date either, so start looking at fares from about six months in advance of your travel date.
'We want to fly as cheap as we can as we will be on a tight budget.' One gets what one pays for: cheap base fares may end up 'costing' more than a bog standard ticket with another airline. Make a few dummy bookings (ie book flights up to the payment page) to see what the full price will be for your journey. Doing this will also answer the 'what cost should we expect to find?' question, with a caveat.
Air fares are very fluid. They are calculated using a number of parameters (number of pax on board, number of pax in each cabin / class of service, number of pax travelling from airports X, Y and Z every day and many more) and go down as well as up the nearer one gets to a travel date. I flew to PHX this month and paid less for the return fare than I did in 2011 and 2012, and I booked about seven weeks before my travel date. Had I booked in June I would have paid more than twice what I did on the day of purchase.
Please book directly with the airline. Many guarantee the lowest fares available with direct bookings. If anything goes amiss you will deal with the airline, not a third party agent (whose responsibility begins and ends with booking flights on your behalf). Please also read the fare rules before purchase, so you will know what you are agreeing to buy.
Some airlines charge for checked baggage, some do not; that information is available at the airlines' websites. There are size and weight limits for checked and hand baggage; airlines tend to be strict about both. The sizes for checked and hand bags includes wheels and handles. Space in overhead lockers is very limited on smaller Airbuses, so please adhere to the rules for hand (carry on) baggage.
You may find that some seats will be offered for free at the time of booking, others will require a payment. You need not pay to book seats, and keep in mind that no airline guarantees that seats selected will be those in which one sits on the day of travel.
Plan to check in on line with the PNR (pax name record, aka locator code) you'll be given on ticket purchase. You may use the PNR to keep an eye on your booking at the airline's website as well. You won't need any paperwork to check in or use at the airport, an e-ticket is just that, stored on the airline's computer - all you will need is identification in order to check in at the airport.
If you do book a flight with a connection, please allow more than the minimum time suggested to make that connection. Expect the unexpected; better to have time at the terminal than to run to the gate and find the fight closing as you reach the desk.
How early to arrive at the airport will to some extent depend on the airline with which you book. If check in and bag drop closes an hour before flight departure, you should plan to be at the airport two hours before flight departure.
http://matrix.itasoftware.com/ is a very useful website to check flight times and fares. Check the websites for Southwest (which has a free checked baggage policy) and jetBlue as well to see what you can create for a DAY to PHX journey.